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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fermat's theorem on sums of two squares ： ウィキペディア英語版 Fermat's theorem on sums of two squares In additive number theory, Pierre de Fermat's theorem on sums of two squares states that an odd prime ''p'' is expressible as :$p\; =\; x^2\; +\; y^2,\backslash ,$ with ''x'' and ''y'' integers, if and only if :$p\; \backslash equiv\; 1\; \backslash pmod.$ For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways: :$5\; =\; 1^2\; +\; 2^2,\; \backslash quad\; 13\; =\; 2^2\; +\; 3^2,\; \backslash quad\; 17\; =\; 1^2\; +\; 4^2,\; \backslash quad\; 29\; =\; 2^2\; +\; 5^2,\; \backslash quad\; 37\; =\; 1^2\; +\; 6^2,\; \backslash quad\; 41\; =\; 4^2\; +\; 5^2.$ On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Albert Girard was the first to make the observation, describing all positive integral numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published posthumously in 1634.〔L. E. Dickson, History of the Theory of Numbers, Vol. II, Ch. VI, p. 227.〕 Fermat was the first to claim a proof of it; he announced this theorem in a letter to Marin Mersenne dated December 25, 1640: for this reason this theorem is sometimes called ''Fermat's Christmas Theorem.'' Since the Brahmagupta–Fibonacci identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer ''n'', we see that if all the prime factors of ''n'' congruent to 3 modulo 4 occur to an even exponent, then ''n'' is expressible as a sum of two squares. The converse also holds.〔For a proof of the converse see for instance 20.1, Theorems 367 and 368, in: G.H. Hardy and E.M. Wright. An introduction to the theory of numbers, Oxford 1938.〕 This equivalence provides the characterization Girard guessed. ==Proofs of Fermat's theorem on sums of two squares== (詳細はEuler after much effort and is based on infinite descent. He announced it in two letters to Goldbach, on May 6, 1747 and on April 12, 1749; he published the detailed proof in two articles (between 1752 and 1755).〔De numerus qui sunt aggregata quorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 4 (1752/3), 1758, 3-40)〕〔Demonstratio theorematis FERMATIANI omnem numerum primum formae 4n+1 esse summam duorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 5 (1754/5), 1760, 3-13)〕 Lagrange gave a proof in 1775 that was based on his study of quadratic forms. This proof was simplified by Gauss in his ''Disquisitiones Arithmeticae'' (art. 182). Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers. There is an elegant proof using Minkowski's theorem about convex sets. Simplifying an earlier short proof due to Heath-Brown (who was inspired by Liouville's idea), Zagier presented a one-sentence proof of Fermat's assertion.〔.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fermat's theorem on sums of two squares」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.